The Sensible Boule Is Not Always Spectacular

The tempting shot is the one that sends the opponent’s boule skittering away. The quieter option is another underarm throw, placed carefully near the jack. I’m collecting pétanque today, and already discovering that spectacle and sensible tactics can point in different directions.

Suppose we give each choice three possible payoffs: −1, 0, or +2 points. These are deliberately simplified estimates of how the choice changes our eventual result, not a complete model of an end. Judging distance, the landing patch, and the consistency of our throw supplies the probabilities.

Pointing has a 10% chance of −1, 65% of zero, and 25% of +2. Shooting offers that attractive +2 outcome 40% of the time, but its −1 risk rises to 45%.

payoff <- c(-1, 0, 2)
point <- c(0.10, 0.65, 0.25)
shoot <- c(0.45, 0.15, 0.40)
expected <- c(point = sum(point * payoff),
              shoot = sum(shoot * payoff))
print(expected)
cat("Choose:", names(which.max(expected)), "\n")

R reports expected values of 0.40 for pointing and 0.35 for shooting. Multiplying each payoff by its probability, then adding, gives a probability-weighted average. The bigger prize happens more often when shooting; the extra failures still outweigh that advantage.

Pascal makes the same arithmetic visible one outcome at a time:

program BouleChoice;
const
  payoff: array[1..3] of real = (-1, 0, 2);
  pointing: array[1..3] of real = (0.10, 0.65, 0.25);
  shooting: array[1..3] of real = (0.45, 0.15, 0.40);
var i: integer; p, s: real;
begin
  p := 0; s := 0;
  for i := 1 to 3 do begin
    p := p + pointing[i] * payoff[i];
    s := s + shooting[i] * payoff[i]
  end;
  writeln('Point: ', p:0:2, ' Shoot: ', s:0:2);
  if p >= s then writeln('Choose: point') else writeln('Choose: shoot')
end.

Neither version promises a good next throw. Expected value describes an average across repeated comparable situations, and our probability estimates may be optimistic. A rough patch of terrain can undo a tidy calculation.

There’s also a tactical wrinkle: maximizing average points needn’t maximize the chance of winning. When only a large gain will help, the spectacular shot may deserve its moment. The arithmetic earns its place by making the chosen objective explicit.